In this paper the $L^1$-stochastic integral and the mixed stochastic integral of a process $Y$ with respect to a process $X$ is defined in a way that extends Riemann-Stieltjes integration of deterministic functions with respect to $X$. The $L^1$-integral will include the classical Ito integral. However, the concepts of "filtration" and adaptability do not play any role; instead, the $p$-variation of Dolean functions of the processes $X$ and $Y$ is the determining factor.
"Stochastic Integration of Processes with Finite Generalized Variations. I." Ann. Probab. 23 (2) 629 - 667, April, 1995. https://doi.org/10.1214/aop/1176988282